Solve (m+n)x^2+2nx=m-n

$$( m + n ) x ^ { 2 } + 2 n x = m - n$$

Solve for m (complex solution)

$$\left\{\begin{matrix}m=-\frac{n\left(x+1\right)}{x-1}\text{, }&x\neq 1\\m\in \mathrm{C}\text{, }&x=-1\text{ or }\left(n=0\text{ and }x=1\right)\end{matrix}\right.$$

Steps for Solving Linear Equation

Use the distributive property to multiply m+n by x^{2}.

$$mx^{2}+nx^{2}+2nx=m-n$$

Subtract m from both sides.

$$mx^{2}+nx^{2}+2nx-m=-n$$

Subtract nx^{2} from both sides.

$$mx^{2}+2nx-m=-n-nx^{2}$$

Subtract 2nx from both sides.

$$mx^{2}-m=-n-nx^{2}-2nx$$

Reorder the terms.

$$mx^{2}-m=-nx^{2}-2nx-n$$

Combine all terms containing m.

$$\left(x^{2}-1\right)m=-nx^{2}-2nx-n$$

Divide both sides by x^{2}-1.

$$\frac{\left(x^{2}-1\right)m}{x^{2}-1}=-\frac{n\left(x+1\right)^{2}}{x^{2}-1}$$

Dividing by x^{2}-1 undoes the multiplication by x^{2}-1.

$$m=-\frac{n\left(x+1\right)^{2}}{x^{2}-1}$$

Divide -n\left(1+x\right)^{2} by x^{2}-1.

$$m=-\frac{n\left(x+1\right)}{x-1}$$

Solve for n (complex solution)

$$\left\{\begin{matrix}n=-\frac{m\left(x-1\right)}{x+1}\text{, }&x\neq -1\\n\in \mathrm{C}\text{, }&x=-1\end{matrix}\right.$$

Steps for Solving Linear Equation

Use the distributive property to multiply m+n by x^{2}.

$$mx^{2}+nx^{2}+2nx=m-n$$

Add n to both sides.

$$mx^{2}+nx^{2}+2nx+n=m$$

Subtract mx^{2} from both sides.

$$nx^{2}+2nx+n=m-mx^{2}$$

Reorder the terms.

$$nx^{2}+2nx+n=-mx^{2}+m$$

Combine all terms containing n.

$$\left(x^{2}+2x+1\right)n=-mx^{2}+m$$

The equation is in standard form.

$$\left(x^{2}+2x+1\right)n=m-mx^{2}$$

Divide both sides by x^{2}+2x+1.

$$\frac{\left(x^{2}+2x+1\right)n}{x^{2}+2x+1}=\frac{m-mx^{2}}{x^{2}+2x+1}$$

Dividing by x^{2}+2x+1 undoes the multiplication by x^{2}+2x+1.

$$n=\frac{m-mx^{2}}{x^{2}+2x+1}$$

Divide -mx^{2}+m by x^{2}+2x+1.

$$n=\frac{m\left(1-x\right)}{x+1}$$

Solve for m

$$\left\{\begin{matrix}m=-\frac{n\left(x+1\right)}{x-1}\text{, }&x\neq 1\\m\in \mathrm{R}\text{, }&x=-1\text{ or }\left(n=0\text{ and }x=1\right)\end{matrix}\right.$$

Steps for Solving Linear Equation

Use the distributive property to multiply m+n by x^{2}.

$$mx^{2}+nx^{2}+2nx=m-n$$

Subtract m from both sides.

$$mx^{2}+nx^{2}+2nx-m=-n$$

Subtract nx^{2} from both sides.

$$mx^{2}+2nx-m=-n-nx^{2}$$

Subtract 2nx from both sides.

$$mx^{2}-m=-n-nx^{2}-2nx$$

Reorder the terms.

$$mx^{2}-m=-nx^{2}-2nx-n$$

Combine all terms containing m.

$$\left(x^{2}-1\right)m=-nx^{2}-2nx-n$$

Divide both sides by x^{2}-1.

$$\frac{\left(x^{2}-1\right)m}{x^{2}-1}=-\frac{n\left(x+1\right)^{2}}{x^{2}-1}$$

Dividing by x^{2}-1 undoes the multiplication by x^{2}-1.

$$m=-\frac{n\left(x+1\right)^{2}}{x^{2}-1}$$

Divide -n\left(1+x\right)^{2} by x^{2}-1.

$$m=-\frac{n\left(x+1\right)}{x-1}$$

Solve for n

$$\left\{\begin{matrix}n=-\frac{m\left(x-1\right)}{x+1}\text{, }&x\neq -1\\n\in \mathrm{R}\text{, }&x=-1\end{matrix}\right.$$

Steps for Solving Linear Equation

Use the distributive property to multiply m+n by x^{2}.

$$mx^{2}+nx^{2}+2nx=m-n$$

Add n to both sides.

$$mx^{2}+nx^{2}+2nx+n=m$$

Subtract mx^{2} from both sides.

$$nx^{2}+2nx+n=m-mx^{2}$$

Reorder the terms.

$$nx^{2}+2nx+n=-mx^{2}+m$$

Combine all terms containing n.

$$\left(x^{2}+2x+1\right)n=-mx^{2}+m$$

The equation is in standard form.

$$\left(x^{2}+2x+1\right)n=m-mx^{2}$$

Divide both sides by x^{2}+2x+1.

$$\frac{\left(x^{2}+2x+1\right)n}{x^{2}+2x+1}=\frac{m-mx^{2}}{x^{2}+2x+1}$$

Dividing by x^{2}+2x+1 undoes the multiplication by x^{2}+2x+1.

$$n=\frac{m-mx^{2}}{x^{2}+2x+1}$$

Divide -mx^{2}+m by x^{2}+2x+1.

$$n=\frac{m\left(1-x\right)}{x+1}$$